2018年IOAA理论第11题-宇宙的热历史

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英文题目

(T11) Thermal History of the Universe (75 points)

Based on Einstein’s general relativity, Russian physicist Alexander Friedmann derived the Friedmann Equation by which the dynamics of a homogeneous and isotropic universe can be well described. TheFriedmann Equation is usually written as follows:

$$(\frac{\dot a}{a})^2=\frac{8\pi G}{3}(\rho_m+\rho_r)+\frac{\Lambda c^2}{3}-\frac{kc^2}{a^2}$$,

We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scale factor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρm isthe density of matter, including dark matter and baryons, ρr is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H0 is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant $$h$$.

(a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time $$t_H$$) using the Hubble parameter. Calculate the present-day Hubble time $$t_{H0}$$.

(b) (5 points) Let us define the critical density ρc as the matter density required to explain the expansion of a flat universe without any radiation or dark energy. Find an expression of the critical density, in terms H and G. Calculate the present critical density ρc0.

(c) (6 points) It is convenient to define all density parameters in a dimensionless manner like $$\Omega_i=\frac{\rho_i}{\rho_c}$$, i.e. the ratio of density to critical density. The Friedmann Equation can be rewritten using these dimensionless density parameters simply as, Ωm + Ωr + ΩΛ + Ωk = 1.

Use this information to find expression for ΩΛ and Ωk, in terms H, c, Λ, k and a.

(d) (7 points) Another equation which is valid for matter, radiation and dark energy is often called the Fluid Equation: $$\dot\rho+3\frac{\dot a}{a}(\rho+\frac{p}{c^2})=0$$, where p is the pressure of some component, ρ is the density and $$\dot\rho$$ is the rate of change of density over time. Radiation contains photons and massless neutrinos, and they both travel atthe speed of light. The pressure exerted by these particles is 1/3 of their energy density. Show that the density of radiation ρr ∝ (1 + z)4, where z is cosmological redshift. You may note that if $$\frac{\dot\rho}{\rho}=n\frac{\dot a}{a}$$, then ρ ∝ an

(e) (4 points) We know that the value of the cosmological constant Λ doesn’t evolve. Its equation of state hasa form p = wρΛc2, where w is an integer. Find the value of w.

(f) (13 points) Planck time, defines a characteristic timescale before which our present physical laws are nolonger valid, and where quantum gravity is needed. The expression for Planck time can be written in terms of $$\hbar$$, G and c and non-dimensional coefficient of this expression in SI units is of the order of unity. Using dimensional analysis, find expression for Planck time and estimate its value.

(g) (7 points) Planck length defines the length scale associated with Planck time is given by lp=ctp. Theminimal mass of a black hole, also called Planck mass, is defined as the mass of a black hole whose Schwarzschild radius is two times the Planck length.

Derive the Planck mass Mp and calculate Mpc2 in GeV. This mass is considered to be an upper threshold for elementary particles, beyond which they will collapse to a black hole.

(h) (4 points) At the very beginning (soon after the Planck time), all the particles were in thermal equilibriumin a primordial soup. As temperature decreased, different particles then decoupled from the primordial soup one by one and could travel freely in the Universe. Photons decoupled at ~300000 years after the Big Bang.These photons emitted at that time are what constitutes the cosmic microwave background (CMB), which follows the Stefan-Boltzmann law for blackbody radiation.

$$\epsilon_r=\frac{\pi^2}{15\hbar^3c^3}(k_BT)^4$$,

Show that the temperature of the CMB follows T/(1 + z) = constant.

(i) (16 points) With the expansion of the Universe, radiation density dropped more quickly than matter density, and at some epoch the matter density was equal to the radiation density. Radiation contains both photons and neutrinos. Apart from photons, neutrinos additionally contribute to the radiation energy densityby 68% (i.e. = 1.68Ωγ0, where γ indicates photons). Estimate the redshift of matter-radiation equality zeq in terms of Ωm0 and reduced Hubble parameter $$h=\frac{H_0}{100kms^{-1}Mpc^{-1}}$$ You may use the current temperature of the CMB: T0 = 2.73K.

(j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe was around 1 MeV. At this time, the radiation density in the universe was much more than all other components. Estimate the time ($$t=\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.

中文翻译

(T11) 宇宙的热历史 (75 points) 基于爱因斯坦的广义相对论,俄罗斯物理学家亚历山大·弗里德曼提出的弗里德曼方程可以很好地描述均匀和各向同性的宇宙的动力学演化。弗里德曼方程经常写成如下的形式:

$$(\frac{\dot a}{a})^2=\frac{8\pi G}{3}(\rho_m+\rho_r)+\frac{\Lambda c^2}{3}-\frac{kc^2}{a^2}$$,

我们将哈勃常数定义为等于$$\frac{\dot a}{a}$$,a是尺度因子,而$$\dot a$$是尺度因子随时间的变化率。于是,哈勃常数可以看做是宇宙时间的函数。在弗里德曼方程中,ρm是物质的密度,包括暗物质和重子物质,ρr是辐射的密度,Λ是宇宙学长度,k是宇宙的曲率。下标0代表宇宙现在的物理量,比如H0就是现在的哈勃常数。另外,为了避免与哈勃常数混淆,我们使用约化普朗克常数$$\hbar=h/2\pi$$而不是普朗克常数h。

(a) (5分)哈勃常数的量纲是什么?使用哈勃常数可以定义出一个描述宇宙膨胀的时标,这就是哈勃时间tH。计算现在的哈勃时间。

(b) (5分)定义临界密度ρc为不考虑辐射和暗能量的平直宇宙保持膨胀时对物质密度做出的限制。使用H和G表示临界密度。计算现在的临界密度ρc0

(c) (6分)使用形如$$\Omega_i=\frac{\rho_i}{\rho_c}$$的方式定义一些无量纲量,代表某个成分的密度占临界密度的比例。使用这些无量纲的密度参数可以将弗里德曼方程改写成很简单的形式,Ωm + Ωr + ΩΛ + Ωk = 1。

利用上述信息,用H, c, Λ, k 和a表示ΩΛ和Ωk

(待续)